Solving Systems of Polynomial

Current geometric and solid modeling systems use semi-algebraic sets for deening the boundaries of solid objects, curves and surfaces, geometric constraints with mating relationship in a mechanical assembly, physical contacts between objects, collision detection. It turns out that performing many of the geometric operations on the solid boundaries or interacting with geometric constraints is reduced to nding common solutions of the polynomial equations. Current algorithms in the literature based on symbolic, numeric and geometric methods suuer from robustness, accuracy or eeciency problems or are limited to a class of problems only. In this paper we present algorithms based on multipolynomial resultants and matrix computations for solving polynomial systems. These algorithms are based on the linear algebra formulation of resultants of equations and in many cases there is an elegant relationship between the matrix structures and the geometric formulation. The resulting algorithm involves singular value decompositions, eigendecompositions, Gauss elimination etc. In the context of oating point computation their numerical accuracy is well understood. We also present techniques to make use of the structure of the matrices to improve the performance of the resulting algorithm and highlight the performance of the algorithms on diierent examples.